Mathematics for the interested outsider

Modules of Generalized Young Tableaux

We can obviously create vector spaces out of generalized Young tableaux. Given the collection of tableaux of shape and content , we get the vector space . We want to turn this into an -module.

First, given any tabloid of shape , we can product a (generalized) tableau by defining to be the number of the row in that contains the entry . As an example, consider the tabloid

This gives us the function , , and . If and we use the usual reference tableau , this gives us the generalized tabloid

The shape of is obviously , and it’s easy to see that the content is exactly . Indeed, there are entries in with the value , just as there are entries in the first row of .

It should also be clear that this correspondence is a bijection. That is, given any generalized tableau of shape and content we can get a tabloid of shape by turning into a function and then putting on row of if .

That means that the basis of generalized tableaux of the vector space is in bijection with the basis of -tabloids of the vector space. And this space carries an action of — the linear extension of the action on tabloids. We want to pull this action across the bijection we just set up to get an action on .

On the one hand, this is as easy as saying it: if corresponds to , we define to be the generalized tableau corresponding to and we’re done. To be a bit more explicit, we define by considering it as a function and setting

So, for example, if

then we can calculate

Even more explicitly, if

then we calculate

We should be clear about a major distinction here: the permutation acts on the entries in — replacing by — but it acts on the places in — moving to the position of .

If we write the correspondence as , then for to be an intertwinor we need . This forces

and so this explicit action is forced on us.

The really interesting thing is that when we use this action on the generalized tableaux in , we always get a module , no matter what shape we start with.

About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.